Conditional Probability (Foundation)
Conditional probability is the probability of an event happening given that another event has already happened, which can change the number of possible outcomes left to consider. It often appears in problems about selecting items without replacement, or when reading two-way tables, where the given condition narrows down the group you are looking at.
Before you start
Make sure you're comfortable with these topics first:
Method
- Identify the 'given' event first; this tells you which reduced group of outcomes you should now be considering.
- If using a table, look only at the row or column matching the given condition, and use its total as the new total.
- If picking items without replacement, reduce both the number of successful outcomes and the total remaining items after each pick.
- Write the probability as (outcomes satisfying both conditions) divided by (outcomes satisfying the given condition).
- Simplify the resulting fraction where possible.
- For a sequence of events, multiply probabilities along the branches of a tree diagram, remembering the second branch depends on the first outcome.
Worked example
A bag contains 5 red counters and 3 blue counters. Two counters are taken from the bag at random, without replacement. Find the probability that the second counter is blue given that the first counter taken was red.
- Since the first counter removed was red, the bag now contains 4 red and 3 blue counters, 7 counters in total remaining.
- The probability the second counter is blue given the first was red is the number of blue counters remaining divided by the total remaining: 3/7.
- So the final answer is P(second blue given first red) = 3/7.
Practice questions
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Q1A box has 4 chocolate and 6 plain biscuits. One biscuit is eaten (it was chocolate), then a second is taken at random. Find P(second is chocolate given first was chocolate).Show answer
Answer: 3/9 = 1/3 (3 chocolate biscuits left out of 9 remaining).
Q2A class has 12 boys and 8 girls. One student, a girl, leaves the room. A second student is then chosen at random from those remaining. Find P(second student chosen is a girl given the first was a girl).Show answer
Answer: 7/19 (7 girls left out of 19 students remaining).
Q3In a survey of 40 students, 25 like Maths. Of those 25, 18 also like Science. Find P(likes Science given likes Maths).Show answer
Answer: 18/25 = 0.72.
Q4A drawer has 7 black socks and 5 white socks. Two socks are taken without replacement. Find P(second sock is black given the first sock was black).Show answer
Answer: 6/11 (6 black left out of 11 remaining).
Q5A pack of 10 cards is numbered 1 to 10. One card is drawn and not replaced; it is even. A second card is then drawn. Find P(second card is even given the first card was even).Show answer
Answer: 4/9 (4 evens left out of 9 cards remaining).
Q6A bag has 3 red, 4 green and 5 blue balls (12 total). One ball is removed at random and is blue. A second ball is then removed. Find P(second ball is red given the first ball was blue).Show answer
Answer: 3/11 (3 red balls still in the bag out of 11 remaining).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A bag contains 6 yellow and 4 purple beads. Two beads are picked at random without replacement. Find the probability that the second bead is purple given that the first bead picked was purple.
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A school careers survey of 90 students found that 54 study a language and 36 do not. Of the 54 who study a language, 42 also do a sport. A student is picked at random from those who study a language. Find the probability that this student also does a sport, giving your answer as a fraction in its simplest form.
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A drawer contains 8 pens: 5 blue and 3 black. Two pens are taken at random, one after another, without replacement. (a) Find the probability that the second pen is black given that the first pen taken was black. (b) Find the probability that both pens taken are black.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the conditional probability (foundation) worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 15 of GCSE Statistics Foundation Workbook 2, the whole course as one free printable PDF.
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